Surplus, Profit and Conditional Expectation

This paper presents an approach to the allocation of surplus and profit loads to different product lines or policies within a single insurance company. The method presented calculates the allocation for each line based on the expected value conditional on the full surplus being exhausted. The use of conditional expectations produces a familiar variance-based allocation for a number of specific cases, and variance is the least squares approximation in the general case. The formula is easily extended to include correlations between policies. It is the intention of this paper merely to highlight the mathematical connection between conditional expectations and variances. Setting profit loads based on variance is already a popular and practical technique, and this additional theoretical support should encourage actuaries to keep it as one method in their "toolbox".